This follows from Deligne's work on differential equations
with regular singularities [1]} and is explained in
[2]}, following work of Malgrange
[3]}, [4]}.
In particular, [3]} explains the case where \(C\)  has
genus zero, and the general case is similar.
We now recapitulate the proof.
The restriction
of \(({E}, \nabla )\)  to \(C\setminus D\)  is a flat vector bundle and hence
gives rise to a locally constant sheaf of \(\mathbb {C}\) -vector spaces
\(\mathbb {V}:=\ker (\nabla )\) 
on \(C\setminus D\) . As \(\Delta \)  is contractible, the
inclusion
\(C\setminus D\hookrightarrow {C}\setminus {D}\) 
is a
homotopy equivalence; thus \(\mathbb {V}\)  extends uniquely (up to canonical
isomorphism) to a local system \(\widetilde{\mathbb {V}}\)  on \({C}\setminus {D}\) .
Manin's local results on extending flat vector bundles across divisors
[1]} imply that there is a
canonical extension, which is unique, up to unique isomorphism,
\((\widetilde{\nabla }:{E}\rightarrow {E}\otimes \Omega ^1_{{C}}(\log {D}))\) 
of
\((\widetilde{\mathbb {V}}\otimes _{\mathbb {C}}{O}_{{C}\setminus {D}}, \operatorname{id}\otimes d)\) 
to a flat vector bundle on \({C}\)  with
regular singularities along \({D}\) , equipped with an isomorphism
\(({{E}}, \widetilde{\nabla })|_C\simeq (E,\nabla ).\)
